Question

follow the seven step strategy to graph the following rational function,
$f(x) = -\frac{2}{x^2 - 9}$

find the horizontal asymptote(s). select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the equation(s) of the horizontal asymptote(s) is/are $y = 0$.
(type an equation. use a comma to separate answers as needed.)
b. there is no horizontal asymptote.

plot points between and beyond each x - intercept and vertical asymptote. find the value of the function at the given value of $x$.
$x$: -8, -4, 0, 4, 8
$f(x) = -\frac{2}{x^2 - 9}$
(simplify your answers.)

Explanation:

Step1: Calculate for x = -8

Substitute \( x = -8 \) into \( f(x) = -\frac{2}{x^2 - 9} \). First, calculate \( x^2 - 9 = (-8)^2 - 9 = 64 - 9 = 55 \). Then \( f(-8) = -\frac{2}{55} \).

Step2: Calculate for x = -4

Substitute \( x = -4 \) into the function. \( x^2 - 9 = (-4)^2 - 9 = 16 - 9 = 7 \). So \( f(-4) = -\frac{2}{7} \).

Step3: Calculate for x = 0

Substitute \( x = 0 \). \( x^2 - 9 = 0 - 9 = -9 \). Then \( f(0) = -\frac{2}{-9} = \frac{2}{9} \).

Step4: Calculate for x = 4

Substitute \( x = 4 \). \( x^2 - 9 = 16 - 9 = 7 \). So \( f(4) = -\frac{2}{7} \).

Step5: Calculate for x = 8

Substitute \( x = 8 \). \( x^2 - 9 = 64 - 9 = 55 \). Then \( f(8) = -\frac{2}{55} \).

Answer:

For \( x = -8 \): \(-\frac{2}{55}\)
For \( x = -4 \): \(-\frac{2}{7}\)
For \( x = 0 \): \(\frac{2}{9}\)
For \( x = 4 \): \(-\frac{2}{7}\)
For \( x = 8 \): \(-\frac{2}{55}\)